A "how many blocks puzzle" answer comes from counting occupied x, y, z coordinates, not from counting visible faces or top diamonds, because some cubes in an isometric pile are drawn with zero faces yet still occupy positions inside supported columns and must be included in your total. In the Counting Blocks Puzzle, the camera faces the positive x, positive y, and positive z sides, and the renderer checks the three neighboring lattice positions in those directions for every cube. A top face is drawn only when the coordinate directly above is empty, one side face appears when the positive x neighbor is empty, and the other side face appears when the positive y neighbor is empty. A cube is completely hidden only when all three of those neighbor positions are occupied, which is why scanning every z layer for filled cells gives the correct total even when the isometric surface shows nothing at all for certain interior cubes. Four number options appear for each scene, an exact selection awards 200 points, and five exact selections produce the maximum 1,000.
What the Isometric Drawing Shows and What It Omits
The visible image in the Counting Blocks Puzzle is only the surface of a larger voxel structure. Each cube sits on an integer lattice at one coordinate written as x, y, z, where the x and y values locate its column on the floor and z gives its height. Because every coordinate above z zero has the coordinate directly below it, the piles contain no floating blocks; every cube sits on top of another cube or on the ground plane.
The renderer applies three independent checks to each cube. If the cell directly above is empty, a top face is drawn. If the positive x neighbor is empty, one side face is drawn. If the positive y neighbor is empty, the other side face is drawn. None of those checks is conditional on the others, so a single cube can show one, two, or three faces depending on which directions are open. A tall column has exactly one top diamond even when it contains five stacked cubes, which is the first place casual counters go wrong.
The Three-Face Rule Behind Every Hidden Cube
A cube disappears from the drawing entirely when all three of its camera-facing neighbors are occupied at the same time. One neighboring cube is not enough to hide an entire cube, and two neighbors are not enough either. If any of the three camera-facing directions is open, at least one face remains visible. Only the three-neighbor case removes the cube from the rendered surface.
| Positive x neighbor | Positive y neighbor | Positive z neighbor | Faces drawn |
|---|---|---|---|
| Empty | Empty | Empty | Three faces (top, x side, y side) |
| Empty | Empty | Occupied | Two faces (x side, y side) |
| Occupied | Occupied | Empty | One face (top only) |
| Occupied | Occupied | Occupied | Zero faces — cube is fully hidden |
The face rule prevents a common counting error: a single shaded side or a single shared top does not mean a cube is missing from the pile. The cube is missing only when you cannot see any face at all. The covered faces are simply omitted from the SVG while the hidden cube remains part of the literal pile.
How to Read the Layer Scanner from Bottom to Top
The practical method for solving the puzzle is to combine the isometric surface with the layer scanner beneath it. The scanner separates the pile into horizontal z slices from bottom to top. Every filled scanner cell represents one occupied x and y coordinate in that layer, including a cube that has no visible face in the isometric view.
The two-dimensional projection constants never change the cube count. The game draws a cube with screen x equal to 150 plus 30 times x minus y, and screen y equal to 80 plus 17 times x plus y minus 34 times z. Those constants control only where each face is drawn on the canvas and never alter the literal voxel set, so you can safely ignore them when you are counting.
- Open the layer scanner beneath the isometric pile and identify the camera direction from the visible top, x side, and y side faces.
- Read the z layers from bottom to top. Each layer is a two-dimensional grid of x and y coordinates.
- Mark every filled scanner cell as one occupied x and y pair at that height, including cells that correspond to a cube with zero visible faces in the isometric view.
- Tally the filled cells for the first layer and keep a running count across all subsequent layers.
- Sum the running counts at the end. The final number equals the number of occupied lattice coordinates in the pile.
- Pick the matching value from the four plain number totals offered for the round.
Empty scanner cells preserve the footprint alignment without contributing a cube, so do not include them in your tally. Reading one layer at a time is what makes the count robust against hidden cubes: even when the isometric surface shows nothing in a region, a filled scanner cell in that region still means one cube is present at that height.
Worked Example: A 2x2 Pile With One Hidden Cube
Consider a 2x2 footprint where the four column heights are 2, 2, 1, and 1. The sum of column heights gives the total cube count directly: 2 + 2 + 1 + 1 = 6 cubes.
The lowest cube at position (0, 0, 0) is the hidden one. Its positive x neighbor at (1, 0, 0) is occupied because that column has height 2, its positive y neighbor at (0, 1, 0) is occupied because that column has height 1, and its positive z neighbor at (0, 0, 1) is occupied because that column has height 2. All three camera-facing neighbors are occupied, so the renderer omits every face for this cube and it appears nowhere in the isometric drawing. The four number options offered for the round would include 6 as the single matching total.
If you counted only the top diamonds, you would see one top per column and report 4 cubes. If you counted visible faces, you would overcount, because a single cube can contribute multiple faces while a hidden cube contributes none. The layer scanner catches the missing cube because the (0, 0) cell is filled at z = 0 in the bottom layer, while the isometric drawing shows no face for that coordinate.
Picking the Right Total From the Four Number Options
Each round shows one original isometric pile and four plain number options. There is no named answer, total, visible count, hidden count, semantic correct marker, or result data attribute stored anywhere in the scene. The four options are plain numbers, and the engine independently counts the literal voxel list to find the single matching option. Choosing that option awards 200 points and immediately presents the next pile.
The five scenes progress from a compact 2x2 platform to a raised ridge, a crowned footprint, diagonal towers, and a broad hidden terrace. Each scene uses transparent geometry rather than a hand-authored answer flag, and the published projection explanation plus the complete layer scan make every occupied coordinate auditable without the engine printing the final sum.
Scoring, Repair, and the Maximum 1,000-Point Run
| Action | Effect on score and state |
|---|---|
| Select the matching total on the first try | +200 points, advance to the next scene |
| Select a wrong total, then select the same wrong total again | Deduplicated; the second identical pick does not consume another miss |
| Select one wrong total, then select the correct total on the same scene | Failure history is cleared before the next scene; no contamination of later rounds |
| Select a second genuinely different wrong total | Run is locked; only Restart remains available |
| Complete five scenes with exact totals | Final score is exactly 1,000 points |
| Press Restart at any point | Scene returns to the exact opening state |
Pointer and touch input select one of four native answer buttons whose targets are taller than 44 pixels. Keyboard play begins by focusing an option; arrow keys move through the two-by-two option grid, and the real DOM focus follows the visible cursor. Enter or Space chooses the current state cursor through a functional update, avoiding a stale answer after fast navigation. When the run is complete or deadlocked, every answer is disabled, while Restart remains available through the shared GameShell.
Common Counting Errors and Why They Fail
The most frequent wrong answer is the number of outlined cube tops. A tall column has one top diamond regardless of how many cubes it contains, so counting only top diamonds reports the column count rather than the cube count. The second most frequent error is counting visible faces. A single cube can contribute a top face and two side faces, another may contribute only one side face, and a fully hidden cube contributes no polygon at all, so the visible-face count is not the cube count.
For the deeper strategy, see the companion walkthrough on how to solve a counting blocks puzzle with hidden cubes, which builds on the same three-face rule with additional practice scenes. The correct quantity in every case is the number of occupied lattice coordinates, equivalent to the sum of all complete column heights, which is exactly what the layer scanner reports one cell at a time.